While studying some paper, I am curious about the following solution. The probability density function is given as below. ($\sigma$, and $\alpha$ can be treated as constants. $X,Y$ are column vectors with size N)
$$L(X,Y|v,\gamma) = \frac{1}{\{(2\pi)^2\sigma^2[\sigma^2+\alpha^4\frac{v}{2}(1+\gamma^2)]\}^N} \exp\left\{−\frac{1}{2\sigma^2[\sigma^2+\alpha^4\frac{v}{2}(1+\gamma^2)]}[(\sigma^2+\alpha^4\frac{v}{2}\gamma^2)X^HX-2\gamma\alpha^4\frac{v}{2}Re\{X^HY\}+(\sigma^2+\alpha^4\frac{v}{2})Y^HY]\right\}.$$
To get the maximum likelihood estimate of $\gamma$, we solve the following partial derivatives. $$\frac{\partial logL}{\partial \gamma}=0, \frac{\partial logL}{\partial v}=0,$$ The papar says that we get the following quadratic equation. $$\gamma^2Re\{X^HY\} - \gamma Y^HY + \gamma X^HX - Re\{X^HY\}=0$$ I tried to get this final quadratic equation, but I couldn't. I only got a fifth degree equation. Maybe something is wrong. Does anybody help me?